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Properties

Label 1489.539
Modulus $1489$
Conductor $1489$
Order $124$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1489, base_ring=CyclotomicField(124)) M = H._module chi = DirichletCharacter(H, M([77]))
 
Copy content gp:[g,chi] = znchar(Mod(539, 1489))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1489.539");
 

Basic properties

Modulus: \(1489\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1489\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(124\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1489.n

\(\chi_{1489}(64,\cdot)\) \(\chi_{1489}(80,\cdot)\) \(\chi_{1489}(91,\cdot)\) \(\chi_{1489}(96,\cdot)\) \(\chi_{1489}(100,\cdot)\) \(\chi_{1489}(120,\cdot)\) \(\chi_{1489}(121,\cdot)\) \(\chi_{1489}(125,\cdot)\) \(\chi_{1489}(134,\cdot)\) \(\chi_{1489}(137,\cdot)\) \(\chi_{1489}(144,\cdot)\) \(\chi_{1489}(150,\cdot)\) \(\chi_{1489}(180,\cdot)\) \(\chi_{1489}(201,\cdot)\) \(\chi_{1489}(216,\cdot)\) \(\chi_{1489}(221,\cdot)\) \(\chi_{1489}(270,\cdot)\) \(\chi_{1489}(324,\cdot)\) \(\chi_{1489}(349,\cdot)\) \(\chi_{1489}(405,\cdot)\) \(\chi_{1489}(407,\cdot)\) \(\chi_{1489}(413,\cdot)\) \(\chi_{1489}(443,\cdot)\) \(\chi_{1489}(486,\cdot)\) \(\chi_{1489}(539,\cdot)\) \(\chi_{1489}(557,\cdot)\) \(\chi_{1489}(563,\cdot)\) \(\chi_{1489}(577,\cdot)\) \(\chi_{1489}(608,\cdot)\) \(\chi_{1489}(729,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{124})$
Fixed field: Number field defined by a degree 124 polynomial (not computed)

Values on generators

\(14\) → \(e\left(\frac{77}{124}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1489 }(539, a) \) \(1\)\(1\)\(e\left(\frac{51}{62}\right)\)\(e\left(\frac{61}{62}\right)\)\(e\left(\frac{20}{31}\right)\)\(e\left(\frac{16}{31}\right)\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{99}{124}\right)\)\(e\left(\frac{29}{62}\right)\)\(e\left(\frac{30}{31}\right)\)\(e\left(\frac{21}{62}\right)\)\(e\left(\frac{11}{62}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 1489 }(539,a) \;\) at \(\;a = \) e.g. 2